Abstract
The modular vector field of a Poisson–Nijenhuis Lie algebroid A is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian A-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson–Nijenhuis structure.
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Agrotis, M., Damianou, P.A.: The modular hierarchy of the Toda lattice. Diff. Geom. Appl. (To appear)
Bangoura M., Kosmann-Schwarzbach Y. (1998). Equation de Yang-Baxter dynamique classique et algebroïdes de Lie. C. R. Acad. Sci. Paris 327(8): 541–546
Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Mathematics Lectures, vol. 10, American Math. Soc., Providence (1999)
Evens S., Lu J.-H., Weinstein A. (1999). Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Quart. J. Math. Oxford 50(2): 417–436
Fernandes R.L. (1993). On the mastersymmetries and bi-Hamiltonian structure of the Toda lattice. J. Phys. A 26: 3797–3803
Fernandes R.L. (2002). Lie algebroids, holonomy and characteristic classes. Adv. Math. 170: 119–179
Fernandes, R.L., Damianou, P.: Integrable hierarchies and the modular class. Preprint math. DG/0607784
Fernandes R.L., Vanhaecke P. (2001). Hyperelliptic Prym varieties and integrable systems. Commun. Math. Phys. 221: 169–196
Flaschka H. (1974). The Toda lattice. I. Existence of integrals. Phys. Rev. B 9(3): 1924–1925
Grabowski J., Marmo G., Michor P. (2006). Homology and modular classes of Lie algebroids. Ann. Inst. Fourier 56: 69–83
Grabowski J., Urbanski P. (1997). Lie algebroids and Poisson-Nijenhuis structures. Rep. Math. Phys. 40(2): 195–208
Grabowski J., Urbanski P. (1997). Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids. Ann. Glob. Anal. Geom. 15: 447–486
Kosmann-Schwarzbach Y. (1996). The Lie bialgebroid of a Poisson-Nijenhuis manifold. Lett. Math. Phys. 38(4): 421–428
Kosmann-Schwarzbach, Y., Magri, F.: On the modular classes of Poisson–Nijenhuis manifolds. Preprint math. SG/0611202
Kosmann-Schwarzbach Y., Magri F. (1990). Poisson-Nijenhuis structures. Ann. Inst. Henri Poincaré 53: 35–81
Kosmann-Schwarzbach Y., Weinstein A. (2005). Relative modular classes of Lie algebroids. C. R. Math. Acad. Sci. Paris 341(8): 509–514
Magri, F., Casati, P., Falqui, G., Pedroni, M.: Eight lectures on integrable systems. In: Kosmann-Schwarzbach, Y. et al. (eds.) Integrability of Nonlinear Systems. Lecture Notes in Physics, 2nd edn. vol. 495, pp 209–250 (2004)
Weinstein A. (1997). The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23(3–4): 379–394
Xu P. (2003). Dirac submanifolds and Poisson involutions. Ann. Sci. École Norm. Sup. 36(4): 403–430
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Caseiro, R. Modular Classes of Poisson–Nijenhuis Lie Algebroids. Lett Math Phys 80, 223–238 (2007). https://doi.org/10.1007/s11005-007-0164-0
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DOI: https://doi.org/10.1007/s11005-007-0164-0